CHAP, xxiv] INTEGERS WITH EQUAL SUMS OF LIKE POWERS. 709 



where x, y, z are not a permutation of u, v, w. Then x+y+z=Q and the 

 equation given by n = 4 is a consequence of the others. Replacing z by 

 xy and w by uv, we get 



Let Xi, 2/1, MI, PI be one solution; the general solution is 



y=P 1 



where x 2 , y 2 , Uz, v 2 are arbitrary. Various special solutions of (4) are given. 

 A. Gerardin 25 noted that 



(f-2g) k +(4j-g) k +(3g-5f) k =(4f-3g) k +(2g-5fi k +(f+g) k (k = l, 2, 4). 



He 26 gave 2d+Zx, 4d+2x, d = d+2x, 4d+3x, 2d. 

 Welsch 27 stated that the general solution of (2) is 



y n -z=- 



x n = t-CD, yn.^t+BD, y n = t-AC, 

 with x i} yi (i = l, , n 3) arbitrary [false if n>3, since in 

 only the terms free of the x's and y's cancel]. 



E. N. Barisien 28 gave the relations involving 1, , 32: 



2 



1, 8, 10, 15, 20, 21, 27, 30=4, 5, 11, 14, 17, 24, 26, 31 



= 2, 7, 9, 16, 19, 22, 28, 29 = 3, 6, 12, 13, 18, 23, 25, 32. 



C. Bisman 29 gave six relations like the last, a numerical example of 

 2a k ='Zb k (k = l, '--, n) for each n^9, and three identities of the type 



n 



a b, a 2c, a+6-f-c, a+2b c = a-\-2b, a+c, a b c, a+b 2c. 



L. Aubry 30 treated 2x i ='2Ui, ^x\ = Lu\ (i=l, 2, 3) by setting X{ = 

 ',-n, whence S7/, = 2y,-. The cubic equation holds if 



m 



E. B. Escott 31 applied his 16 method to the last problem. 



A. de Farkas 32 noted that, if 2z, Zz 2 , 2x 3 and x 3 +3a* 4 H ----- \-(ml)x 

 equal the analogous sums involving y's, then Xi+a, x 2 +a+d, -, x m -\-a 

 + (m l)d have the same sum and sum of cubes as y\-\-a, [false]. 



G. Tarry 33 stated that the first 2 n (2o+l) integers can be separated into 

 two sets each of 2 n-1 (2a+l) integers having the same sum of fth powers for 

 * = 1, -, n. For o = l, n = 3, the first set is 1, 3, 7, 8, 9, 11, 14, 16, 17, 18, 

 22, 24. __ 



K Assoc. fran. av. sc., 39, I, 1910, 44; Sphinx-Oedipe, 5, 1910, 182. 



26 Sphinx-Oedipe, 5, 1910, 177. 



27 L'interme'diaire des math., 18, 1911, 60 (for n = 3), 205. 



28 Mathesis, (4), 1, 1911, 69. 



29 Ibid., 205-8, 264. 



80 L'interme'diaire des math., 19, 1912, 156-7. E. Miot (p. 3) gave two numerical solutions. 



41 Ibid., 263-4. 



12 Ibid., 182. His remark on p. 131 is the case n=2 of Frolov'e 7 first result. 



** Ibid., 200. 



